This justifies, in the above mentioned regimes, methods in which an implicit coupling
between the pressure gradient and the energy flux is enforced. This strategy has been proposed in the seminal paper [1]} and in the more recent works [2]}, [3]}.
We finally assume that the only acting volumetric force is gravity, so that
\(\mathbf {f} =-g\mathbf {k}, \)  where \(g\)  denotes the acceleration of gravity and \( \mathbf {k} \) 
the upward pointing unit vector in the standard Cartesian reference frame.
It follows that
\(\frac{\mathcal {T}}{{\mathcal {U}}} \rho g &=& \frac{g{\mathcal {T}} {\mathcal {U}}}{{\mathcal {U}^2}} \rho = \frac{g{\mathcal {L}}}{{\mathcal {U}^2}} \rho =\frac{\rho }{Fr^2} \ \ \ \ Fr^2=\frac{{\mathcal {U}^2}}{g{\mathcal {L}}} \nonumber \\\frac{\mathcal {L}}{{\mathcal {E}}}\rho g&=&\frac{g \rho {\mathcal {L}}}{{\mathcal {I}}+\frac{\mathcal {P}}{\mathcal {R}} + {\mathcal {U}}^2} =\frac{g \rho {\mathcal {L}}}{{\mathcal {U}^2}}\frac{1}{\frac{\mathcal {I}}{\mathcal {U}^2} + 1 + \frac{1}{Ma^2}} =\frac{ \rho }{Fr^2}O(Ma^2).\) 
